Exact solution for the bending deformations of layered ...Exact solution for the bending deformations of layered magneto-electro-elastic laminates based on thin-plate formulation 8

International Journal of Engineering and Applied Sciences (IJEAS)

ISSN: 2394-3661, Volume-3, Issue-4, April 2016

7 www.ijeas.org

Abstract—In this paper, a rather compact differential

equation governing the bending behavior of a

magneto-electro-elastic (MEE) rectangular thin plate is

introduced, in particular, the exact solutions for the deformation

response of laminated BaTiO3-CoFe2O4 composites subjected

to certain types of surface loads are analytically obtained. Due to

the omission of the transverse shear deformation and rotatory

inertia assumed in Kirchhoff thin-plate theory, the governing

equation can accordingly be expressed in terms of the transverse

displacement only. As a result, the structural characteristics

such as elastic displacements, electric potential and magnetic

induction for a magneto-electro-elastic (MEE) rectangular plate

can be carried out in a theoretical approach. For a laminate

MEE composite, the material constants can be uniquely

determined by the volume-fraction (v.f.) of the piezoelectric

constituent BaTiO3, and are tabulated with 25% offset of the

volume-fraction. According to the specified boundary conditions

imposed on the MEE thin plate, the deformation variations with

closed-circuit electric restriction are evaluated analytically in

the present study. The results obtained in this paper by using the

proposed model can be shown to have good agreements with the

other available research works, however, with the advantage

that the present study indeed provides a much simpler way in

seeking the analytic solutions for the interactively coupled

quantities of a layered MEE medium.

Index Terms—Layered structures, Mechanical properties,

Analytical modelling

I. INTRODUCTION

It is well-known that structure made of piezoelectric

materials can produce voltage when external stress is applied,

nevertheless, it will also induce stress when voltage difference

is implemented across the structure. Parallel to the

piezoelectricity, piezomagnetizm can intrinsically be

characterized by interactive couplings between system's

magnetic polarization and mechanical strain, i.e., in a

piezomagnetic medium, one may induce a spontaneous

magnetic moment by applying physical stress, or reversely

obtain a physical deformation by applying a magnetic field.

Composites structures consist of layered piezoelectric and

piezomagnetic components, which can be referred as one kind

of the magneto-electro-elastic (MEE) material, possesses the

ability to convert energy of magnetism, electricity or elasticity

into another form. The MEE material also exhibits a specific

magneto-electric effect which is not appeared in a

single-phase piezoelectric or piezomagnetic material.

Surprisingly, in some cases the magneto-electric effect of

MEE composites can even be obtained two orders larger than

Mei-Feng Liu, Dept. of Financial and Computational Mathematics,

I-Shou University, Kaohsiung, Taiwan R.O.C, +886-6577711ext.5625,

that of a single-phase magneto-electric material with highest

magneto-electro coefficient [1]. Due to these multiphase

properties, the MEE material have been found increasing

applications in making efficient smart and intelligent

structures such as magnetic field probes, electric packing,

acoustic, hydrophones, medical ultrasonic imaging and so on.

For the past few years, systematic investigations either in

determining the material coefficient of such new type material

or in analyzing the static or dynamic behavior under certain

external conditions are vigorously proposed by many

professional engineers and scientists. Li [2] studied the

average magneto-electro-elastic fields in the multi-inclusion

embedded in an infinite matrix, and estimated the effective

magneto-electro-elastic moduli of piezoelectric-

piezomagnetic composites for both BaTiO3 fiber reinforced

CoFe2O4 and BaTiO3-CoFe2O4 laminate. It has been shown

that the magneto-electro coupling demonstrated by

magneto-electro coefficients vary with the volume faction of

BaTiO3 and have opposite signs for fibrous and laminated

composites, respectively. Pan [3] obtained the exact solution

for three dimensional, anisotropic, linear

magneto-electro-elastic, simply-supported and multilayered

rectangular plates under static loadings. It is stated that even

for relatively thin plate, responses from an internal load are

quite different to those from a surface load. Chen et. al. [4]

establish a micro-mechanical model to evaluate the effective

properties of layered magneto-electro-elastic composites and

the linear coupling effect between elasticity, electricity and

magnetism of the MEE composite is accordingly derived. In

their study, numerical results for a BaTiO3-CoFe2O4

composite with 2-2connectitivi are obtained, and the

dependences of magneto-electro-elastic coefficients, the

so-called product properties, of the composite on the volume

fraction of BaTiO3 are clearly depicted. Wang and Shen [5]

extended their previous works on piezoelectric media to study

the general solution of three-dimensional problems in

transversely isotropic magneto-electro-elastic media through

five newly introduced potential functions. Chen and Lee [6]

simplified the governing equations of the linear theory for the

magneto-electro-thermal-elastic plate with transverse

isotropy by introducing two displacement functions and stress

functions. In their study, two new state space equations are

established while selecting certain physical quantities as the

basic unknowns. Wang et. al. [7] performed a state vector

formulation for the three dimensional, orthotropic and

linearly magneto-electro-elastic multiple layered plate and

expressed the basic unknowns by collecting not only the

displacement, electric potential and magnetic potential but

also some of the stresses, electric displacements, and

magnetic induction. A boundary integral formulation for the

plane problem of magneto-electro-elastic media are

Exact solution for the bending deformations of

layered magneto-electro-elastic laminates based on

thin-plate formulation

Mei-Feng Liu

Exact solution for the bending deformations of layered magneto-electro-elastic laminates based on thin-plate

formulation

8 www.ijeas.org

performed by Ding and Jiang [8] using strict differential

operator theory. They obtained the fundamental solutions for

an infinite MEE plane in terms of four harmonic functions

which satisfied a set of reduced second order partial

differential equation for distinct eigenvalues case. Guan and

He [9] derived the fundamental equation for the plane

problem of a transversely isotropic magneto-electro-elastic

media by applying the Almansi’s theorem and expressed all

physical quantities by four harmonic functions for distinct and

non-distinct cases.

In this study, a rather simple analytic solution for the

deformations of the magneto-electro-elastic (MEE)

rectangular plate under certain type of applied loads acting on

the top surfaces are derived. By imposing the Kirchhoff

thin-plate hypothesis on the plate constituent, the governing

equation in terms of only the transverse displacement of the

plate can be obtained and therefore a rather compact form

indicating the multiple effects between elasticity, electricity

and magnetism of the plate can be successfully presented. The

MEE plate is chose to be made of the two-layered

BaTiO3-CoFe2O4 laminate, which can be thought as a

transversely isotropic magneto-electro-elastic medium and

the material coefficients for such continuum can be expressed

uniquely by introducing the volume-fraction of BaTiO3 in the

layered composite. The corresponding deformation analysis

regarding the elastic displacements, electric potential and

magnetic induction of the MEE thin plate is evaluated through

the formulation mentioned in this study. Some comparisons

with previous literatures are made and great agreements are

reached which directly validate the proposed simplification

for the MEE modeling.

II. FORMULATIONS

For a Magneto-Electro-Elastic (MEE) thin plate made of

two-layered BaTiO3-CoFe2O4 laminate, the fundamental

Kirchhoff hypothesis for the small-deflection of simple

bending problem can be applied, and thus the transverse shear

deformations and rotary inertias can be neglected, also the

transverse shear strains are negligible. In accordance with the

assumptions that in-plane electric fields and magnetic fields in

a very thin medium can be ignored [10], that is, only the

transverse electric, 3E , and magnetic field, 3H , are under

consideration, the following governing equation for the

bending problem can be found [11]

4 4 4 ( , )D w E w M w P x y , (1)

where 3

11

12

c hD represents the plate rigidity,

331 1

12

e hE

denotes the effective rigidities induced by

electricity, 3

31 2

12

q hM

is the corresponding effective

rigidities cause by magnetism and ( , )P x y denotes the

applied load acting on the top surface. Herein h is the plate

thickness, 233 33 33d , 1 31 31 33 33( )e d q and

2 33 31 33 31( )q d e are extra-defined parameters aiming

to simplify the derivations; ijc , ije , ijq , ijd and ij are the

elastic, dielectric, piezoelectric, piezomagnetic,

magnetoelectric, and magnetic constants, respectively. It

should be noted the relation 11 12 662c c c for transversely

isotropic material is adopted in the above equation. As it can

be also learned from Ref. [11], the reduced extended traction

vectors can be stated as follows,

2 2

11 12 31 312 2

2 2

12 11 31 312 2

2 2 2

66 662

x

y

xy

w wc z c z e q

z zx y

w wc z c z e q

z zx y

w w wc z z c z

x y y x x y

, (2)

31 31 33 33z

u vD e e d

x y z z

, (3)

31 31 33 33z

u vB q q

x y z z

. (4)

Meanwhile, the following expression for the electric potential

and magnetic potential can be derived,

211( , )z w x y

z

, (5)

221( , )z w x y

z

, (6)

where 1( , )x y and ( , )1 x y represent the variations of

electric field and magnetic field in the thickness direction

while the plate is under deformation, and are both

independent of z variable.

In seeking for the solution to Eq. (1), we can assume the

following expression for the transverse deflection of the MEE

plate,

1 1

( , ) ( ) ( )mn m n

m n

w x y A X x Y y

, (7)

where ( )mX x and ( )nY y are the homogeneous solutions of

Eq. (1) and can be determined according to the assigned

boundary conditions. Some mode shapes and corresponding

eigenvalues with respect to commonly seen boundary

conditions are tabulated in Table 1 as a reference. It should be

noted that the mode shapes ( )mX x and ( )nY y not only

satisfy the corresponding boundary conditions but also

possess the intrinsic orthogonality, i.e., we will have

0

( ) ,( ) ( ) ( )

0,

xLm

m M m mM

X x if m MX x X x dx X x

if m M

, (8)

and

0

( ) ,( ) ( ) ( )

0,

yLn

n N n nN

Y y if n NY y Y y dy Y y

if n N

, (9)

where xL and yL denotes the plate lengths along x and y

directions. After ( )mX x and ( )nY y are determined, we can

further expand the applied load on the top surface of the plate

into the generalized double Fourier series as

1 1

( , ) ( ) ( )mn m n

m n

P x y p X x Y y

, (10)

in which the Fourier coefficient mnp can be determined as

follows



9 www.ijeas.org

0 0

1( , ) ( ) ( )

( ) ( )

x yL L

mn m nm n

p P x y X x Y y dxdyX x Y y

. (11)

By substituting Eq. (7) and Eq. (10) into Eq. (1) and

collecting the constant term, we can have the following

equation

4 2 2 42m m n n mn mnD E M A p , (12)

where m and n are the corresponding eigenvalues

associated to the specific boundary conditions. Furthermore,

the magnitude of the transverse deflection, mnA , can be

determined, i.e.,

4 2 2 42

mnmn

m m n n

pA

D E M

. (13)

Thus we have the exact solution for the transverse deflection

due to applied load

4 2 2 41 1

( , ) ( ) ( )2

mnm n

m n m m n n

pw x y X x Y y

D E M

, (14)

and the exact solution for the mechanical displacements along

x- and y- directions can be expressed as

4 2 2 41 1

-( ) ( )

2

mnm n

m n m m n n

pwu z zX x Y y

x D E M

, (15)

4 2 2 41 1

-( ) ( )

2

mnm n

m n m m n n

pwv z zX x Y y

y D E M

(16)

Table 1: Mode shapes and the corresponding eigenvalues for

specified boundary conditions

As for the electric boundary conditions and magneto

boundary conditions specified on the top and bottom surfaces

of the plate, there are normally two kinds of cases to be

discussed.

Case I: Closed-circuit, i.e., ( , , / 2) 0x y h and

( , , / 2) 0x y h

By carrying out the anti-derivatives for Eq. (5) and Eq. (6), we

can have the following expressions for electric potential and

magnetic potential as

2 211 0( , , ) ( , ) ( , ) ( , )

2x y z z w x y z x y x y

(17)

2 221 0( , , ) ( , ) ( , ) ( , )

2x y z z w x y z x y x y

(18)

since ( , , / 2) 0 x y h , after solving the above two

equations, one can get

1( , ) 0x y and 2 210 ( , ) ( / 2) ( , )

2x y h w x y

,

thus the exact solution for the electric potential due to applied

load can be expressed as

2 2 2 21 1( , , ) ( , ) ( / 2) ( , )2 2

x y z z w x y h w x y

. (19)

By the same token, we can have the following exact solution

for the magnetic potential due to applied load as

2 2 2 22 2( , , ) ( , ) ( / 2) ( , )2 2

x y z z w x y h w x y

. (20)

Case II: Open-circuit, i.e., ( , , / 2) 0zD x y h and

( , , / 2) 0zB x y h

Substituting Eqs. (5)-(6) and Eqs. (15)-(16) into Eqs. (3) and

(4), the solutions for the electric displacement and magnetic

induction in thickness direction can be expressed as

21 231 33 33 33 1 33 1- - ( , ) - ( , )zD e d z w x y d x y

, (21)

21 231 33 33 33 1 33 1- - ( , ) - ( , )zB q z w x y x y

(22)

therefore, the open-circuit conditions will lead to

21 231 33 33 33 1 33 1( , , / 2) / 2 - ( , ) - ( , ) 0zD x y h e d h w x y d x y

(23)

21 231 33 33 33 1 33 1( , , - / 2) - / 2 - ( , ) - ( , ) 0zD x y h e d h w x y d x y

(24)

and

21 231 33 33 33 1 33 1( , , / 2) ( / 2) - ( , ) - ( , ) 0zB x y h q d h w d x y x y

(25)

21 231 33 33 33 1 33 1( , , - / 2) - ( / 2) - ( , ) - ( , ) 0zB x y h q d h w d x y x y

. (26)

In viewing of Eq. (23) through Eq. (26), it can be found that

these four equations cannot provide any information about

determining the expressions for 1( , )x y and 1( , )x y , nor

for the terms ( , )0 x y and 0 ( , )x y . This is because the

variations of electric displacement and magnetic

displacement along thickness direction are vanished under the

proposed methodology, i.e., 0z zD B

z z

, therefore it is

expected to see the linear dependency of these two quantities

on the z variable. In that sense, open-circuit boundary will

result in a trivial solution with the displacement components

approaching to zero and makes the expressions of 1( , )x y

and 1( , )x y being negligible as can be detected from Eqs.


formulation

10 www.ijeas.org

(21) and (22). As a result, two more boundary conditions will

be needed in order to resolve the expressions for the terms

( , )0 x y and 0 ( , )x y . For that reason, in this study, we

don’t specifically focus on the open-circuit case and only

adopt the closed-circuit boundary condition as our numerical

examples discussed in the next section.

It should be noted that the in-plane electric fields and

magnetic fields can be ignored if the plate thickness is very

small (e.g. 10 xh L ), and only the transverse electric field,

3E , and magnetic field, 3H , are related to the electric

potential and magnetic potential in the follow form

according to the Maxwell’s equations.

211- - ( , )zE z w x y

z

, (27)

221- ( , )zH z w x y

z

, (28)

in which the terms 1( , )x y and 1( , )x y can be determined

by considering the closed-circuit case on the plate surfaces.

By substituting Eqs. (5)-(7) into Eq. (2), the stress

distributions are respectively

11 12

1 231 31

31 1 31 1

( , , ) - ( ) ( ) - ( ) ( )

- ( ) ( ) ( ) ( )

( , ) ( , )

x mn m n mn m n

mn m n m n

x y z c A zX x Y y c A zX x Y y

e q A z X x Y y X x Y y

e x y q x y

, (29)

12 11

1 231 31

31 1 31 1

( , , ) - ( ) ( ) - ( ) ( )

- ( ) ( ) ( ) ( )

( , ) ( , )

y mn m n mn m n

mn m n m n

x y z c A zX x Y y c A zX x Y y

e q A z X x Y y X x Y y

e x y q x y

, (30)

66-2 ( ) ( )xy mn m nc A zX x Y y . (31)

The above equations provide the analytic solutions to the

physical quantities of a magneto-electro-elastic thin plate

subjected to surface applied load and can be decided

according to the transverse deflection ( , )w x y and potential

parameters ( , , )x y z and ( , , )x y z .

III. NUMERICAL DEMONSTRATIONS

In this section, some examples based on the proposed

model for the thin magneto-electro-elastic rectangular plate

subjected to external loads and closed-circuit electric

restrictions are presented. For the first a few examples, the

author demonstrates some special cases which have been

verified in previous research works in order to validate the

present study and tell the differences with other available

literatures.

Mode shape validation. Since the mode shape

corresponding to the free vibration of the MEE plate is of

interest due to its importance on the nature and extent of the

electro-magneto-mechanical coupling, we first check the

validation of fundamental mode shapes for a piezoelectric

square plate made of commonly used PZT4 material by

utilizing the present model. In order to compare the results

with those proposed by Heyliger & Saravanos [13], the

dimensions of the plate are chosen to be / 01xL h , x yL L

and 0.01 mh , and the boundary conditions are assumed to

be simply-supported around four edges with closed-circuit on

the top and bottom surfaces, i.e.

( , , - / 2) ( , , - / 2) 0x y H x y H . The thickness is

normalized as /Z z H , the displacements and electric

potential are normalized with respects to their own largest

values along thickness direction, i.e., / max( )W w w ,

/ max( )U u u and / max ( )u . The normalized

distributions for both the transverse and in-plane mode shapes

along with the electric potential for a PZT4 square thin plate

are depicted in Fig. 1, as we can detect from this figure, it is

almost identical to Fig. 2(a) in the paper conducted by

Heyliger & Saravanos [13]. Therefore, good agreement has

been reached.

Fig 1. Thickness distributions for a PZT4 thin plate with

dimension / 01xL h , x yL L and 0.01 mh subjected to

simply-supported and closed-circuit boundary conditions.

Example 1. The second example is an electro-elastic single

layer rectangular plate made of purely piezoelectric BaTiO3.

The length-to-thickness ratio and the width-to-thickness ratio

are chosen to be / 10 xL H and / 5 yL H in order to

match the dimension setting in the paper presented by Chen

and Lee [6] although this doesn’t assure the thin-plate

requirement to be satisfied. All the surfaces of the plate are

assumed to be traction free except the bottom surface, on

which a sinusoidal loading

0 sin( / )sin( / )x yP n x L m y L with amplitude

20 1 / N m and mode number 1m n is applied. In

particular, the deformation responses of the plate are

calculated at the fixed horizontal coordinates

0 0( , ) (0.75 ,0.25 )x yx y L L . Fig. 2(a)-(f) indicates the

variations along thickness direction of the elastic

displacements w , xu , yu , electric potential ( , , )x y z ,

magnetic potential ( , , )x y z , electric displacement zD and

magnetic displacement zB for such a plate caused by a

sinusoidal loading on the top surface with all edges

simply-supported. As we can recognize from these figures,

the shear deformation is linearly dependent on the transverse

deformation and the electric potential reveals quadratic

variation along z coordinate. These two phenomena can be

also found in both the papers studied by Pan [3] and Chen [6],

see Fig. 1 and Fig. 2 in their studies respectively. It should be

noted that the electric potential and magnetic potential in this

case are identically zero due to the fact that 0z zD B

z z



11 www.ijeas.org

and close-circuit electric restriction is adopted in this paper

instead of open-circuit one.

Fig. 2. Variations along thickness direction of the elastic

displacements, electric potential, magnetic potential, electric

displacement and magnetic displacement for a single layer

BaTiO3 plate caused by sinusoidal loading on the top surface

with all edges simply-supported.

Table 2. Material constants for both laminate and fibrous

MEE multiphase composites, partially cited from Buchanan

[12]

Example 2. In this example, a two-layered

magneto-electro-elastic plate made of equally-placed

BaTiO3-CoFe2O4 laminate is then presented. The

piezoelectric BaTiO3 is placed on the bottom layer

( / 2 0H z ) whereas the magnetostrictive CoFe2O4 is on

the top layer ( 0 / 2z H ), the material constants for this

kind of MEE plate can be found in Table 2 with the

volume-fraction set to be 50% for the lamination case. The

dimensions are 1000 mxL , 20 myL , 1 mh ,

/ 20 yS L h , 1n and the related physical quantities are

normalized as follows with all of them still remain

dimensional, 3/u S , 3/v S , 4100/w S , 2/xx S , 2/yy S ,

2/xy S , 2/ S ,

2/ S , /zD S , and /zB S . The

deformation variations of the magneto-electro-elastic plate

due to external load, 0( , ) sin / yP x y P y L , applied on the

top surface with magnitude 0 1 2 N/mP are presented in Fig.

3 at the location 0 0( , ) ( / 2, / 4)xx y L S . Fig. 3(a)-(h) are the

variations of the elastic displacements components ( , )w x y

and 0 0( , , )yu x y z , electric potential 0 0( , , )x y z , magnetic

potential 0 0( , , )x y z , electric displacement zD , magnetic

displacement zB , the normal stress components

11 0 0( , , )x y z and 2 0 0( , , )2 x y z along the thickness

direction with boundary condition to be simply-supported

around four edges. From these figures, we can obviously

observe the interactive behavior between the piezoelectric,

piezomagnetic and magnetoelectric effects for a

magneto-electro-elastic (MEE) plate under mechanical

applied load. If the MEE plate is assumed to be very thin, the

deformation variations for both the electric and magnetic

potentials reveal a quadratic dependence on the thickness

variable, however, for the other related quantities such as

shear deformations, electric and magnetic displacements as

well as the stress distributions, linear dependence can be still

detected.

From now on, the bi-layered BaTiO3-CoFe2O4 laminate

working as a magneto-electro-elastic plate will be discussed

based on the Kirchoff hypothesis and the thin-plate theory.

The simplified governing equation, Equation (1), will be

examined subjected to various kinds of surface applied load,

and the deformation behavior with different boundary

conditions imposed on the plate will also be inspected. Due to

the simplicity of the proposed model, the higher mode

response can be easily performed; therefore, the deformation

variation of the MEE plate with respect to different mode

numbers will be carried out as a reference.

The dimensions of the MEE plate is set to be

1 *1 *0.05m m mx yL L H unless otherwise mentioned,

however, any kind of dimensions can be applied as long as the

span-to-thickness ratio is satisfying the requirement for

thin-plate theory, i.e., / 10xL H . The plate surfaces are

assumed to be traction free except on the top or bottom

surface, on which a z-direction surface load is applied. The

external load can be of any type possibly occurs in the study

of MEE plate, however, in order to observe the variation of

the deformations, three commonly seen static forces are

performed in this paper. They are

(1) uniform load, i.e., 0( , ) P x y P ,

(2) distributed load, i.e., 0( , ) ( ) ( ) M NP x y P X x Y y and

(3) concentrated load, i.e., 0 0 0( , ) ( - , - ) P x y P x x y y .


formulation

12 www.ijeas.org

Fig. 3. Variations of the elastic displacements, electric

potential, magnetic potential, electric displacement, magnetic

displacement and normal stress components along the

thickness direction for a two-layered magneto-electro-elastic

BaTio3-CoFe2O4 plate caused by external loading on the top

surface with all edges simply-supported.

The geometric boundary conditions imposed on the MEE

plate can be any one of the combinations for free, clamped

and simply-supported edges; however, in this study, only the

boundary conditions of simply-supported around, clamped

around and cantilever plates are conducted as numerical

examples. It should be noted that the deformation responses

for the MEE plate under static loads will be calculated at a

fixed horizontal coordinate 0 0( , )x y , however, due to the

different plate characteristic with respect to different

boundary conditions, the horizontal coordinates will be

various according to the corresponding boundary conditions.

That is, for SSSS plate location is chosen to be at

0 0( , ) (0.5 ,0.5 )x yx y L L , for CCCC plate at

( , ) (0.5 ,0.5 )0 0 x yx y L L and for CFFF plate at

0 0( , ) ( ,0.5 )x yx y L L .

Fig. 4 (a)-(d) are the deformation variation of electric

potential, magnetic potential, electric displacement and

magnetic displacement for the laminated BaTiO3-CoFe2O4

MEE plate with simply-supported around edges and surface

distributed load sin( / )sin( / )x yP x L y L with respect

to various volume fraction of BaTiO3. As it can be observed,

the concavity for the pure piezoelectric BaTiO3 plate

(v.f.=100%) is uniquely different from the other MEE

laminates with a much higher magnitude. Although the

dependence with respect to z variable remains quadratic, the

electric potential for the MEE lamination with volume

fraction other than 100% reveals a negative sign along the

thickness direction instead of a positive one in the pure

BaTiO3 case. However, the magnitudes placed in order are

25%, 75% and 50%, 0% for the pure piezomagnetic CoFe2O4

case stands zero with no piezoelectric effect due to the applied

load, which is quite reasonable. As for the magnetic

potentials, quadratic relation can also be verified and the

concavity for each case seems to be consistent. The maximum

magnitude occurs in the 25% laminate followed by the 75%,

50% and 0% cases, 100% for pure piezoelectric BaTiO3 case

stays nil with piezomagnetic effect being vanished.

Fig. 4-1. Variation of (a) electric potential (b) magnetic

potential for the lamination MEE plate with simply-supported

BCs under surface distributed load.



13 www.ijeas.org

Fig. 4-2. Variation of (c) electric displacement (d) magnetic

displacement for the lamination MEE plate with

simply-supported BCs under surface distributed load.

For the MEE laminate presented in Fig. 4, the electric and

magnetic displacements, also called fluxes, can be found to be

linearly dependent on the z variable and the slope varies

slightly with respect to the volume fraction of the MEE plate.

However, if we near watch the magnitudes, we can find that

all of them are pretty slim. This is due to the nature of the

material parameter itself, also because of the close-circuit

restriction we impose on the model. Nevertheless, if the

electric and magnetic boundary conditions are chosen

otherwise, the magnitudes for electric and magnetic

displacements may be changed in a different way.

Whatsoever, in the present study, only the close-circuit MEE

thin plate is considered, therefore the author hereafter leave

out all the presentations and discussions for the electric

displacement and the magnetic displacement due to their

insignificant effects.

Since the deformation variations for the laminate MEE

plates with clamped-around edges and under uniform applied

load 21 N/mP are quite similar to those behaviors

presented in Fig. 4, the author therefore skip these two case

and directly go to the cases for the cantilever laminate and

fibrous MEE plate. Also the deformation variations for the

electric and magnetic displacements are neglected due to their

small magnitudes both approaching to zero.

Fig. 5 (a)-(b) are the deformation variation of electric

potential, magnetic potential versus volume fraction for a

cantilever laminated BaTiO3-CoFe2O4 plate subjected to unit

impulse force acting on the location 0 0( , )x y , i.e.,

0 01 ( - , - ) P x x y y . The deformation behavior of a

cantilever MEE plate is quite similar to the corresponding one

of a simply-supported MEE plate except for the concavity.

Owing to the different characteristics of MEE plate with

different boundary conditions and subjected to different types

of applied load, the sign change on the concavity is reasonable

and expected.

Fig. 5. (a)-(b) Deformation variation of electric potential and

magnetic potential versus volume fraction for a cantilever

laminated BaTiO3-CoFe2O4 square plate subjected to unit

impulse force acting on the location 0 0( , )x y , i.e.,

0 01 ( - , - ) P x x y y .

The deformation variation of electric potential and

magnetic potential for the 50% laminated BaTiO3-CoFe2O4

MEE plates subjected to various loads with all edges

simply-supported are demonstrated in Fig. 6 (a)-(b). And the

corresponding deformation variations for the laminate MEE

composites with all edges clamped are depicted in Fig. 7

(a)-(b). Followed by Fig. 8 (a)-(b), the same illustrations for

the MEE cantilever plates are provided. As we can see from

these figures, the concentrated applied load always stimulate

much stronger deformation as we expected and followed by

the distributed applied load, uniform applied load seems to be


formulation

14 www.ijeas.org

less operative.



MEE plate subjected to various loads with all edges

simply-supported.



MEE plate subjected to various loads with all edges

simply-supported.



MEE cantilever plate subjected to various loads.

Finally, in order to see the influence of mode orders, the

deformation variation of electric potential and magnetic

potential versus mode number for the MEE laminate under

certain applied load are given in Fig. 9 (a)-(b) with all edges

simply-supported, in Fig. 10 (a)-(b) with all edges clamped,

and in Fig. 11 (a)-(b) for the cantilever plate. It should be

noted that some of the mode orders contributes nothing on the

plate deformations because of maybe the symmetry of the

mode shape or the location we calculate at.


magnetic potential versus mode number for the 50%

laminated BaTiO3-CoFe2O4 MEE simply-supported plate

subjected to sinusoidal loading.

Fig. 10. (a)-(b) Deformation variation of electric potential

and magnetic potential versus mode number for the 50%

laminated BaTiO3-CoFe2O4 MEE clamped plate subjected to

sinusoidal loading.



15 www.ijeas.org

Fig. 11. (a)-(b) Deformation variation of electric potential

and magnetic potential versus mode number for the 50%

laminated BaTiO3-CoFe2O4 MEE cantilever plate subjected

to sinusoidal loading.

IV. CONCLUSION

The closed form solutions for the bending problem of a

bi-layered BaTiO3-CoFe2O4 composite are derived based on a

new invented governing equation for magneto-electro-elastic

(MEE) rectangular thin plate, in particular, the elastic

displacements, electric potential and magnetic induction for a

magneto-electro-elastic (MEE) laminate are implemented

analytically.

It has been shown that the material coefficients for the

MEE constituent vary a lot according to the volume-fraction

of BaTiO3 it contains. The deformation variations for the

MEE thin plate with closed-circuit electric restriction are

evaluated with respect to various boundary conditions, and

the effects of the volume-fractions are investigated in detail. It

can be found that the shear deformation is linearly dependent

on the transverse deformation, whereas the electric and

magnetic potential are both of quadratic variation along the

thickness direction. In addition, the deformation behavior for

a single phase material can be found to be quite different from

the multiphase one in either the magnitude or the sign it is

induced by.

The present study provides some commonly seen examples

for the magneto-electro-elastic (MEE) rectangular thin plate

under the action of 3 kinds applied loads, and offers the

discrepancy on the deformation variation of electric potential

and magnetic induction with respect to various typical

boundary conditions. This work proposed a much easier and

systematic way to seek for the analytic solutions for the

deformation characteristics of a bi-layered MEE thin plate,

and should be of interest to someone devoted on the practice

of structure design with the fully coupled medium.

ACKNOWLEDGMENT

This work was partially supported by the National Science

Council of the Republic of China under Grant

NSC-95-2211-E327-046. The author is grateful for this

support.

REFERENCES

[1] Van Run AMJG, Terrel DR, Scholing JH. An in situ grown eutectic

magnetoelectric composite material. Journal of Material Science 1974;

9(10):1710-1714.

[2] Li JY. Magnetoelectroelastic multi-inclusion and inhomogeneity and

their applications in composite materials. International Journal of

Engineering Science 2000; 38:1993-2011.

[3] Pan E. Exact solution for simply supported and multilayered

magneto-electro-elastic plates. Transaction of the ASME Journal of

Applied Mechanics 2001; 68:608-618.

[4] Chen ZR, Yu SW, Meng L, Lin Y. Effective properties of layered

magneto-electro-elastic composites. Composite Structures 2002;

57:177-182.

[5] Wang X, Shen YP. The general solution of the three-dimensional

problems in magnetoelectroelastic media. International Journal of

Engineering Science 2002; 40:1069-1080.

[6] Chen WQ, Lee KY. Alternative state space formulations for

magnetoelectric thermalelasticity with transverse isotropy and the

application to bending analysis of non-homogeneous plates.

International Journal of Solids and Structures 2003; 40:5689-5705.

[7] Wang JG, Chen LF, Fang SS. State vector approach to analysis of

multilayered magneto-electro-elastic plates. International Journal of

Solids and Structures 2003; 40:1669-1680.

[8] Ding HJ, Jiang AM. A boundary integral formulation and solution for

2D problems in magneto-electro-elastic media. Computers and

Structures 2004; 82:1599-1607.

[9] Guan Q, He SR. Two-dimensional analysis of

piezoelectric/piezomagntic and elastic media. Composite structures

2005; 69:229-237.

[10] Tzou HS. Piezoelectric Shells, Distributed Sensing and Control of

Continua. Netherlands: Kluwer Academic Publishers, 1993.

[11] Liu MF, Chang TP. Closed form expression for the vibration problem

of a transversely isotropic magneto-electro-elastic plate. Journal of

Applied Mechanics-Transactions of The ASME 2010; 77:

024502-1-024502-8.

[12] Buchanan GR. Layered versus multiphase magneto-electro-elastic

composites. Composites Part B 2004; 35:413-420.

[13] Heyliger P, Saravanos DA. Exact free-vibration analysis of laminated

plates with embedded piezoelectric layers. The Journal of the

Acoustical Society of America 1995; 98(3): 1547-1557.

Documents

Exact solution for the bending deformations of layered ...Exact solution for the bending deformations of layered magneto-electro-elastic laminates based on thin-plate formulation 8